Properties

Label 4-3240e2-1.1-c1e2-0-5
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 14·19-s + 6·23-s − 7·29-s − 31-s − 4·37-s + 9·41-s + 6·43-s − 2·47-s + 7·49-s − 55-s + 3·59-s + 10·61-s + 2·67-s − 2·71-s − 4·79-s − 6·83-s + 14·89-s + 14·95-s − 2·97-s − 9·101-s + 6·103-s + 4·107-s + 6·109-s − 6·115-s + 11·121-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 3.21·19-s + 1.25·23-s − 1.29·29-s − 0.179·31-s − 0.657·37-s + 1.40·41-s + 0.914·43-s − 0.291·47-s + 49-s − 0.134·55-s + 0.390·59-s + 1.28·61-s + 0.244·67-s − 0.237·71-s − 0.450·79-s − 0.658·83-s + 1.48·89-s + 1.43·95-s − 0.203·97-s − 0.895·101-s + 0.591·103-s + 0.386·107-s + 0.574·109-s − 0.559·115-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258560519\)
\(L(\frac12)\) \(\approx\) \(1.258560519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.886969200731195783610123488199, −8.546427124431478521912065956665, −8.110719199251530693463066107274, −7.72107542061064072730207554392, −7.37386854581611387259722473585, −6.94626203376860517157058496202, −6.58573833081599472373357771697, −6.34748410794748119229715275041, −5.77059179378850071528799696860, −5.50778085643179359860210389790, −4.98969988818598053042667425669, −4.43388319362375098944391287347, −4.13311091801798342121841708923, −3.91854136012179800051379803532, −3.37042191233296151122071031471, −2.73753060388801700091898098854, −2.22644791967771007060631995971, −1.95948385673889945936172138544, −1.11717829738032397418185346735, −0.37161786521798621901467683054, 0.37161786521798621901467683054, 1.11717829738032397418185346735, 1.95948385673889945936172138544, 2.22644791967771007060631995971, 2.73753060388801700091898098854, 3.37042191233296151122071031471, 3.91854136012179800051379803532, 4.13311091801798342121841708923, 4.43388319362375098944391287347, 4.98969988818598053042667425669, 5.50778085643179359860210389790, 5.77059179378850071528799696860, 6.34748410794748119229715275041, 6.58573833081599472373357771697, 6.94626203376860517157058496202, 7.37386854581611387259722473585, 7.72107542061064072730207554392, 8.110719199251530693463066107274, 8.546427124431478521912065956665, 8.886969200731195783610123488199

Graph of the $Z$-function along the critical line